Transition Rate and the Photoelectric Effect in the Presence of a Minimal Length

In this work, according to the generalized uncertainty principle (GUP) and time- dependent perturbation theory, the transition rate in the present of a minimal length based on the Kempf algebra is studied. Also, we find the absorption cross section in the framework of GUP. The modified photoelectric effect is investigated and we show that the differential cross section of photoelectric effect in the frame work of GUP is related to the isotropic minimal length scale. The upper bound on the isotropic minimal length is estimated.


Introduction
In the last decade, different theories of quantum gravity such as the string theory, loop quantum gravity, noncommutative geometry and doubly special relativity are proposed for finding the unification between the general theory of relativity and the standard model of particle physics [1]. Although these theories are different in concepts, all of these studies lead to unique belief which predicts the existence of a measurable minimal length scale. An immediate consequence of existence of a minimal length is that the Heisenberg uncertainty principle is modified. Nowadays the modified uncertainty principle is called generalized uncertainty principle (GUP) [2]. The generalized uncertainty principle corresponding to the modified Heisenberg algebra can be written as △X△P ≥h where β is a positive parameter [3,4] and also, Eq. (1) yields a minimal measurable length (△X) min =h √ β. At this time, many studies have been done to compute the corrections of quantum mechanics in the GUP framework. These investigations seem to modify mechanical Hamiltonians at atomic scales [5][6][7][8][9][10]. In the recent years, a lot of papers have been devoted to the gravity and reformulations of quantum field theory in the presence of a minimal length scale [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Kempf and his collaborators have introduced finite resolution of length can be obtained from the deformed Heisenberg algebra [26][27][28]. The Kempf algebra in a D-dimensional space is characterized by the following deformed commutation relations wherei, j = 1, 2, ..., D and β, β ′ are two positive deformation parameters. In Eq. (2), X i and P i are position and momentum operators in the GUP framework. According to Eq. (2), we can easy to find the following an isotropic minimal length scale It should be mentioned that in a D + 1-dimensional space-time the following Lorentz-covariant deformed algebra have introduced by Quesne and Tkachuk [29,30] [X µ , where µ, ν, ρ = 0, 1, 2, · · · , D and g µν = g µν = diag(1, −1, −1, · · · , −1). In the present work, we study the transition rate and photoelectric effect in the presence of a minimal length. For this purpose, we consider the formalism of time-dependent perturbation theory to the interactions of atomic electron with the modification of classical radiation field. The paper is organized as follows: In Sec. 2, the Hamiltonian of atomic electron is obtained in the presence of a minimal length. From this modified Hamiltonian we find the modified transition rate. Also, we investigate the absorption cross section in the presence of a minimal length. In Sec. 3, the photoelectric effect in the presence of a minimal length is studied. According to this study we obtain the upper bound on the isotropic minimal length. Our conclusions are presented in Sec. 4. We use SI units throughout this paper.

Transition Rate and Absorption Cross section in the Presence of a Minimal length
The purpose of this section is finding the transition rate and an absorption cross section in the presence of a minimal length based on the Kempf algebra. So that we must to introduce the representation of modified position and momentum operators which are satisfied Kempf algebra in Eq. (2). In Ref. [31], Stetsko and Tkachuk introduced the approximate representation fulfilling the Kempf algebra in the first order over the deformation parameters β and β ′ where the operators x i , p i satisfy the canonical commutation relation and p 2 = D i=1 p i p i . It is interesting to note that in the special case of β ′ = 2β, the position operators commute in linear approximation over the deformation parameterβ, i.e. [X i , X j ] = 0. The following representations, which satisfy Kempf algebra in the special case of β ′ = 2β, was introduced by Brau [32]

Interaction of Atomic electron with the Radiation Field in the Presence of a Minimal Length
In this section, let us obtain the time-dependent perturbation theory to the interactions of atomic electron with the classical radiation field. Classical radiation field means that the electric or magnetic field derivable from a classical radiation field [33]. The basic Hamiltonian, with A 2 vanished, is where in the coulomb gauge (∇ · A = 0),we have used A · p = p · A. We work with a monochromatic field of the plane wave for whereǫ andn are the polarization and propagation direction. In Eq. (9),ǫ is perpendicular to the propagation directionn and from Eq. (8) treat − e mec A · p as time-dependent potential. Now, for obtaining the Hamiltonian in the presence of a minimal length, we must replace the usual position and momentum operators with the modified position and momentum operators according to Eqs. (6) and (7), we have After neglecting terms of order β 2 and higher in Eq. (10), the modified Hamiltonian can be obtained as follows From above equation we can consider the following modified time-dependent potential As we know that the exp(−iωt)-term is responsible for absorption, while the exp(iωt) -term is responsible for stimulated emission. Now, let us treat the absorption case in the presence of a minimal length. We have (13) and the modified transition rate is obtained as follows where

Absorption Cross Section in the Presence of a Minimal Length
In this section we want to obtain the absorption cross section in the framework of GUP. An absorption cross section has a definition as follow ( Energy unittime )absorbed by the atom(i → n) Energy f lux of the radiation f ield , where the energy flux of the radiation field is given by In this case we use the following definition for the electromagnetic field Let us obtain the modified energy flux of the radiation field. From Eq. (17), first we must find the classical electromagnetic field in the presence of a minimal length so that we write the electromagnetic field in Eq. (17) by using the modified position and momentum operators which are satisfied the Quesne-Tkachuk algebra, that is where ⊔ ⊓ := ∂ µ ∂ µ is the d'Alembertian operator. If we substitute Eq. (18) into Eq. (17), we will obtain the following modified electromagnetic field

After inserting Eq. (9) into Eq. (19), we have
where k = ω cn . According to Eqs. (16) and (20), we can easy obtain the energy flux in the presence of a minimal length as follows where Now, from Eqs. (14), (15) and (22), we can obtain the modified absorption cross section as follows where (σ abs ) 0 = (σ abs ) 0 = (w i→n ) 0 In the above equation α is e 2 hc .

Photoelectric Effect in the Presence of a Minimal Length Scale
The photoelectric effect means that the ejection of an electron when an atom is placed in the radiation field. The process of photoelectric effect is considered to be the transition from an atomic state to a continuum state [33]. According to previous section, |i is the ket for an atomic state, while |n can be taken to be a plane-wave state |k f . Now, we want to study the modified differential cross section for the photoelectric effect by using our earlier formula for modified absorption cross section. To find the number of states it is convenient to use the box normalization convention for plane-wave states. Assuming a plane-wave state normalized that means if we integrate the square modulus of its wave function for a cubic box of side L, we obtain unity. Also, the state is considered to satisfy the periodic boundary condition with periodicity of the side of the box. In the limit L → ∞, the number of states is reduced to the number of dots in three-dimensional lattice space. If we consider the energy of the final-state plane wave (E =h 2 k 2 f 2me ) and the periodic boundary condition, we can easy find the following number of states in the interval between E and E + dE where dΩ is the solid angle element. According to Eqs. (24) and (25) the modified differential cross section for the photoelectric effect is obtained as follows where By considering the initial-state wave function is the ground-state hydrogen atom wave function, Eqs. (27) and (28) become where a 0 is the Bohr radius. After using the integrating by parts and the perpendicularn toǫ, we will obtain the modified differential cross section for the photoelectric effect as follows where q = (k f − ω cn ) and (ǫ · k f ) 2 = k 2 f sin 2 (θ) cos 2 (ϕ). If we consider the first term ( dσ dΩ ) 0 , the usual differential cross section and the second term is ( dσ dΩ ) modif ied the relative modification of differential cross section can be obtained as follows Another hand, if we substitute β ′ = 2β into Eq. (3), we will obtain the following isotropic minimal length The isotropic minimal length in three spatial dimensions is given by Hence, by inserting Eq. (35) into Eq. (33), we will obtain the relative differential cross section as follows Now we can estimate the upper bound on the isotropic minimal length in modified photoelectric effect. If we consider the value of differential cross section is about 10 −28 m 2 and also considering ω ≈ 10 14 Hz, a 0 ≈ 10 −11 m, K f ≈ 10 18 m −1 . Here, according to Eq. (36), we can estimate the following upper bound on the the isotropic minimal length It is interesting to note that the numerical value of the upper bound on isotropic minimal length in Eq. (37), is close to the Planck length scale(L p ≈ 10 −35 m).

Conclusions
Heisenberg believed that every theory of elementary particles contain a minimal length scale [34,35]. Today we know that every theory of quantum gravity predicts a minimal observable distance. An immediate consequence of GUP is a modification of position and momentum operators. This study has found the transition rate and photoelectric effect in the presence of a minimal length. First, we have considered the time-dependent perturbation theory and then the modified Hamiltonian of atomic electron was obtained up to the first order over the deformation parameter β. According to the modified Hamiltonian the transition rate in the framework of GUP was investigated. We have seen that the modified transition rate was included in two terms, one term was usual transition rate and the second term was its correction due to the considered minimal length effect. Hence, we have assumed the modified cross section in two terms and then two terms of modified cross section have been obtained. It is necessary to note that , in the limit β → 0, the modified cross section become the usual cross section. Also, the photoelectric effect in the presence of a minimal length was investigated. We have shown that the relative differential cross section was related to the isotropic minimal length. The upper bound on the isotropic minimal length scale has been estimated. It is interesting to note that the upper bound on the isotropic minimal length was close to the Planck length scale.